Activity 3: GRAPHS AND EQUATIONS (Work and Kinetic Energy) GRAPHS AND EQUATIONS (USING WORK AND KINETIC ENERGY) OBJECTIVES To apply the rules in plotting the numerical results of an experiment. To interpret the graphs and determine the relationship between two physical quantities such as a. Position and time b. Velocity and time c. Acceleration and time d. Applied force and work done by the applied force e. Work done on the object and the change in kinetic energy Formulate an equation relating the two quantities based on the data and the graphs Theory I. GRAPHS A graphical presentation is often used as an effective tool to show explicitly how one variable varies with another. By plotting the numerical results of an experiment and observing the shape of the resulting graph, a relationship between two quantities can be established. The shape of the graph gives us a clue of the relationship of the variable involved. Some of the common ones are the following: A straight-line graph indicates linear or direct relationship between two quantities. A hyperbolic graph indicates an inverse relationship. A parabolic graph tells us of a specific kind of linear or direct relationship. The specific equation relating the two variables of the graph can only be formulated when the graph is linearized. We will see how this can be done in the succeeding discussion. A. Straight Line Graphs A.1 Linear Relationship Figure 1 shows a straight-line graph that does not pass through the origin. This is a linear graph. it shows a linear relationship between the two variables. It means that there is a first-degree relationship between the Celsius readings and the Fahrenheit readings. The general equation for a linear graph is (Equation 1) SCIENVP/LBYENVP revised Feb. 2012 y = mx + b -1- In general.Activity 3: GRAPHS AND EQUATIONS (Work and Kinetic Energy) where m and b are constants. (2) to Eq. Fahrenheit Reading vs. we get the corresponding value in Celsius which is 17. we get values within the line such as 50oC for the corresponding Fahrenheit reading of 122oF. A. Celsius Reading ---------------------------------------------------------------------------------------------------------------------------------------------- Substituting the value of the slope obtained in Eq. the equation relating them is: (Equation 4) y x y = kx or k=y x where k is the constant of proportionality. If we extend the line downward until the temperature is 0oF.8 or 9/5 ---------------------------------------------------------------------------------------------------------------------------------------------Fahrenheit (oF) 32 68 104 140 176 Celsius (oC) 0 20 40 60 80 Graph #1. the slope can be obtained using Eq. x = 20o. (1) and considering that the y-axis is oF and the x-axis is oC. In this case. (1): (Equation 2) 68o = m(20o) + 32o m = 1. we say that the distance is directly proportional to time. and b = 32o in graph #1. By interpolation.2 Direct Proportionality Figure #2 shows a straight line passing through the origin. SCIENVP/LBYENVP revised Feb. m is the slope of the lien and b is the y-intercept. 2012 -2- . when two variables x and y are directly proportional to each other. This equation shows that the quotient of the two variables is always equal to a constant. If we take y = 68o. When time is doubled the distance is also doubled. the equation relating Fahrenheit reading and Celsius reading is therefore: (Equation 3) o F = (9/5)oC + 32o We can also extrapolate values from the graph.8 oC. The y-intercept of the line is the value of y when x is zero. The zero values for both variables simultaneously occur. a parabolic graph passing through the origin can be obtained for the quantities x and y obeying the following equations: (Equation 6) y = kx2. The value is independent of the choice of scales and it expresses a significant fact about the relationship between the plotted variables. Rewriting Eq. y = kx3. For example.Activity 3: GRAPHS AND EQUATIONS (Work and Kinetic Energy) -----------------------------------------------------------------------------------------------------------------------------------------------------Distance Time (m) (s) 0 0 20 1 40 2 60 3 80 4 100 5 120 6 Graph #2. SCIENVP/LBYENVP revised Feb. Distance vs. B. y = kxn The relationship between x and y can be expressed as y xn. the physical slope represents the constant k: (Equation 5) slope = y = y2 – y1 d x x2 – x1 t The physical slope is always our concern in graphical analysis. Parabolic Graphs In general. (6). the geometrical slope which is defined to be tan . on the choice of scales. y = k(constant) xn the ratio of y and xn is a constant. the slope of the distance vs. (where is the angle between the straight line connecting the points and the x-axis) depends on the inclination of the line and hence. On the other hand. Time Graph ------------------------------------------------------------------------------------------------------------------------------------------------------ In graph #2.….. time graph represents the average speed of the object. y = kx4. 2012 -3- . ” To verify this relationship. (6). the equation is y = k/x SCIENVP/LBYENVP revised Feb.. square of time. 2012 -4- . y 1/xn. y = k/x2. For n = 1. squaring the time in the data yields a direct square relationship between height and time. The result is shown in Graph #4.. C. By inspection.…. “height is directly proportionally to the square of time. The specific equation can be verified by determining the value of n. y = k/x3.e. plot height vs. Hyperbolic Graphs Hyperbolic graphs can be obtained for quantities obeying the following equations: (Equation 7) y = k/x.Activity 3: GRAPHS AND EQUATIONS (Work and Kinetic Energy) ------------------------------------------------------------------------------------------------------------------------------------------------------ Height (y) (m) 0 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 Time(t) (s) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ----------------------------------------------------------------------------------------------------------------------------Graph #3 shows a parabolic graph. Thus we say. y = kxn A hyperbolic graph indicates an inverse relationship between two quantities i. the value of n determines the specific equation relating x and y. From Eq. In equation form (Equation 8) y1 x y=k x or k = xy where k is a constant which is equal to the slope of y vs. 2012 -5- . To linearize it. 1/x yields a straight-line graph as shown in Graph #6.16 Graph #6.Activity 3: GRAPHS AND EQUATIONS (Work and Kinetic Energy) ---------------------------------------------------------------------------------------------------------------------------------------------- (y) 200 100 67 50 40 33 (x) 1 2 3 4 5 6 Graph #5.2 0.33 0. A Hyperbolic Graph --------------------------------------------------------------------------------------------------------------------------------------------------C. y 200 100 67 50 40 33 x 1 2 3 4 5 6 1/x 1 0.5 0. 1/x graph. try n = 1 such that y = 1/x. Hence y is directly proportionally to 1/x or y is inversely proportional to x. A linearized version of Graph #5 SCIENVP/LBYENVP revised Feb.1 Inverse Proportionality Graph #5 shows a hyperbolic graph. Plotting y vs.15 0. 8 3.8 0.4 3. The variable (y) is inversely proportional to the square of x.2 1. This kind of relationship is called inverse square proportionality.8 1.2 2.1 2.7 9.4 0.4 0. 1/x will not yield a straight line but plotting y vs.Activity 3: GRAPHS AND EQUATIONS (Work and Kinetic Energy) C. plotting y vs.6 0.9 4. y 16.4 4. 2012 -6- . Graph #7 illustrates such a case.4 SCIENVP/LBYENVP revised Feb.3 x 0.2 Inverse Square Proportionality Sometimes. 1/x2 will yield one.5 0. a. and v is the speed of the object in m/s2. from Newton’s second law of motion. WORK and KINETIC ENERGY A. M is in kilograms (kg). respectively.Activity 3: GRAPHS AND EQUATIONS (Work and Kinetic Energy) II. the work done (in Joule or J) is given by: (Equation 8) W Fnet d where Fnet is parallel to the motion of the object. Hence. it is said to possess kinetic energy which is defined as the energy of an object by virtue of its motion. KE is in kg m2/s2 which is equivalent to Newton·meter or Joule. If the initial and final speeds of the object at its initial and final positions. KINETIC ENERGY When an object of mass M moves with a speed v. (Equation 9) Fnet Ma B. 2012 -7- .2 m) with end stops and adjustable legs SPARK vue Set of mass Pulley Level indicator SCIENVP/LBYENVP revised Feb. It is given by (Equation 10) KE 1 M 2 2 In equation 10. If the acceleration. are known. WORK For an object with mass M that experiences a constant net force Fnet (in Newtons or N) over a distance d (in meters or m). (in m/s2) of the object is known. then the change in kinetic energy KE can be obtained from (Equation 11) KE KE f KE i Method and Set-up: EQUIPMENT NEEDED Position Sensor Dynamic Cart Dynamic Track (1. that passes over a light frictionless pulley as shown in the figure. b) velocity-time graph. 2012 -8- . kinetic energy. as the cart moves along the track. m. To see the relationship between work done. The following graphs can also be seen from the software: a) position-time graph. placed on the dynamic track is attached by a string to a hanging mass. and work done vs change in kinetic energy. its position. A position sensor is fixed in placed at the other end of the track. the students will graph their experimental results on work done vs applied force. From the initial and final positions and the constant acceleration. the work done using equation 8 can be calculated. And from the initial and final velocities. behind the cart. the dynamic cart. The position sensor is attached to the computer via the PASCO USBlink and using the DATA STUDIO software. with mass M. The experimental Set-up In this experiment. The hanging mass provides the constant pull needed to move the cart. and c) acceleration-time graph. velocity. the change in kinetic energy using equation 11 can be obtained. and applied force. and acceleration is recorded. SCIENVP/LBYENVP revised Feb.Activity 3: GRAPHS AND EQUATIONS (Work and Kinetic Energy) Pulley Dynamic Cart with mass M Weight hanger w/hanging mass m Figure 1. Weigh the dynamic cart and then arrange the equipments as shown in figure 1. Make sure that the dynamic track is horizontally level 2. Use the level indicator to check if the track is leveled.Activity 3: GRAPHS AND EQUATIONS (Work and Kinetic Energy) Procedure: A. 3. This screen shot will appear. Tap open and locate the file “work and kinetic energy”. 2012 -9- . If not. adjust the legs of the track to level it by adjusting its legs. This screen shot will appear. Data Collection 1. Turn on SPARK and connect the position sensor to one of the ports of the SPARK vue. SCIENVP/LBYENVP revised Feb. m. Short discussion about the background of the experiment is provided.10 - . Tap but do not release the cart yet. M. Release the cart after 2 secs. Tap the to take a screen shot of the page. Position the cart to about 20 cm from the position sensor. Repeat this particular procedure three times so that you will have three runs and three graphs. Tap the to take a screen shot of the page. You will be ask to enter the mass of the cart. 2012 . and the hanging mass. 6. If you want to save a particular screen just tap and will automatically be save on your journal. Just tap on the appropriate box to enter the M and m. SCIENVP/LBYENVP revised Feb. The screen shot below will appear. On page 5 which is the start of the data collection. 5. Tap to proceed to the next page. Tap the to navigate through the rest of the file.Activity 3: GRAPHS AND EQUATIONS (Work and Kinetic Energy) 4. Tap the button before the cart reach 75 cm. SCIENVP/LBYENVP revised Feb. On that page you will be able to delete a particular run or if you want all your data runs. Tap manage data. 8. 2012 . and the If you made a mistake and you want to delete a particular run. just tap experiment tools will appear as shown in the figure below.Activity 3: GRAPHS AND EQUATIONS (Work and Kinetic Energy) 7. and then manage runs. Tap to proceed to the analysis page.11 - . A walk through of the procedure and how to calculate the work done and the kinetic energy is provided. Fnet. SCIENVP/LBYENVP revised Feb. you will get the distance covered by the cart from the graph. On page 9. the distance. Do this procedure for all the three runs and write the corresponding answers on the box provided on the right side.Activity 3: GRAPHS AND EQUATIONS (Work and Kinetic Energy) B. Tap to open the tools palette. Calculation of the distance d from the graph. Use the graph tools to find the distance d between the release point of the cart to the cart’s farthest position from the motion sensor. The difference between the two is the distance moved by the cart. Tap the to take a screen shot of the page. i. Choose minimum and maximum. 2012 . The screen shot is given below 10.d.12 - . 9. acting on the cart must be determined. The minimum value is the initial distance and the maximum value is the maximum distance. Tap to open graph statistics. travelled by the cart and the net force. Analysis: Calculation of the work done Work done is given by W Fnet d Where Fnet Ma To calculate the work done. from the graph. Use the graph tools to find the acceleration of the cart from the velocity vs time graph.13 - . You will see that the line within the points will have a SCIENVP/LBYENVP revised Feb. Tap and choose any two points along the diagonal line of the graph. Calculation of the acceleration. Tap to open the tools palette. a. 2012 .Activity 3: GRAPHS AND EQUATIONS (Work and Kinetic Energy) ii. The screen shot is given below 12. time graph. The velocity vs time graph looks like the figure below 13. On page 11. you will get the acceleration of the cart from the graph velocity vs. 11. 14 - SCIENVP/LBYENVP revised Feb. 2012 . Do this procedure for all the three runs and write the corresponding answers on the box provided on the right side. the work done for each run can be determined. Tap to get the slope of the line. Tap the to take a screen shot of the page. 14. From the calculated distance covered and acceleration. On page 12. The slope is given by the quantity m and it represents the acceleration. Analysis: Calculation of the change in kinetic energy The kinetic energy of the cart is given by KE 1 M 2 2 The change in kinetic energy is therefore KE KE f KE i Where KE f 1 2 M f 2 and KE i 1 2 M i 2 . the calculated work done is written C.Activity 3: GRAPHS AND EQUATIONS (Work and Kinetic Energy) yellow highlight. You can now calculate the change in kinetic energy of the cart. maximum. 2012 . Tap the 18. Tap to open the graph tools palette. On the next page you will write your calculated change in kinetic energy. To determine the final velocity of the cart. on page 15. Use the graph tools to find the final velocity of the cart from the velocity vs time graph. the velocity vs time graph of your three runs are presented. What is the initial velocity of the cart? What is the initial kinetic energy of the cart? 16. SCIENVP/LBYENVP revised Feb.15 - . The screen shot of page 15 is shown below 17. Tap to open graph statistics.Activity 3: GRAPHS AND EQUATIONS (Work and Kinetic Energy) 15. The maximum value of the velocity represents the final velocity of the cart Do this procedure for all the three runs and write the corresponding answers on the to take a screen shot of the page. box provided on the right side. d (m) Acceleration.KE . (N) F = Ma Run 1 Run 2 Run 3 Work done. M = ___________ kg Hanging mass. a (m/s2) Net Force. (J) W = F d Initial Velocity (m/s) Initial kinetic energy (J) Final Velocity (m/s) Final kinetic energy (J) Change in Kinetic Energy. x 100 W + KE 2 SCIENVP/LBYENVP revised Feb. KE = 1 M ( 2 i2 ) f 2 Percentage difference between the Work done And change in kinetic energy Percentage difference = W . 2012 .Activity 3: GRAPHS AND EQUATIONS (Work and Kinetic Energy) Names: ______________________________ ______________________________ ______________________________ ______________________________ WORK AND KINETIC ENERGY DATA SHEET TRIAL 1: Mass of the dynamic cart. m = _____________kg Quantities Initial position (m) Final position (m) Distance covered*.16 - . 2012 .Activity 3: GRAPHS AND EQUATIONS (Work and Kinetic Energy) SCIENVP/LBYENVP revised Feb.17 - . m = _____________kg Quantities Initial position (m) Final position (m) Distance covered*. d (m) Acceleration. a (m/s2) Net Force.KE .18 - SCIENVP/LBYENVP revised Feb. KE = 1 M ( 2 i2 ) f 2 Percentage difference between the Work done And change in kinetic energy Percentage difference = W .Activity 3: GRAPHS AND EQUATIONS (Work and Kinetic Energy) Names: ______________________________ ______________________________ ______________________________ ______________________________ WORK AND KINETIC ENERGY DATA SHEET TRIAL 2: Mass of the dynamic cart. x 100 W + KE 2 . (N) F = Ma Run 1 Run 2 Run 3 Work done. (J) W = F d Initial Velocity (m/s) Initial kinetic energy (J) Final Velocity (m/s) Final kinetic energy (J) Change in Kinetic Energy. 2012 . M = ___________ kg Hanging mass. If you increase the hanging mass. Draw the cart and then indicate the forces acting on the cart.19 - . is the work done related to the kinetic energy? Why or why not? SCIENVP/LBYENVP revised Feb. The net force on the cart c. The work done on the cart d. Based from your answer on number 5. Which force/s does/do not do work on the cart? Why? 3. What physical quantity does the slope of the position vs time graph represents? d. What physical quantity does the slope of the velocity vs time graph represents? 4. 2. In this experiment. 2012 . velocity.Activity 3: GRAPHS AND EQUATIONS (Work and Kinetic Energy) QUESTIONS 1. Acceleration of the cart b. The change in kinetic energy of the cart 6. What can you say about the value of the calculated work done and the change in kinetic energy? Are the values close to one another? Why or why not? 5. and acceleration as the cart moves from its initial position to the final position? c. what do you think will happen to the following quantities? a. These are the position vs time graph and velocity vs time graph and acceleration vs time graph. What do these graphs tell you about the carts position. a. What are the shape of the position vs time graph and velocity vs time graph? b. two graphs were recorded.